A transition matrix method for generating near-optimal closed-loop solutions to non-linear zero-sum differential games†

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Abstract
This paper presents a method for generating near optimal closed-loop solutions to zero-sum perfect information differential games. Such a near-optimal solution is generated by periodically updating the solution to the two-point boundary-value problem (TPBVP) obtained from the application of the necessary conditions for a saddle-point solution. This procedure is accomplished by updating the co-state vector at each updating time based on the state error from a reference TPBVP solution. The relationship between the required change in the co-state vector and the state error is obtained using the transition matrices for the linearized TPBVP. Between updating times the player using this method plays his open-loop control determined from the updated TPBVP solution. A number of examples are presented to illustrate the advantages and shortcomings of this method.

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